1. INTRODUCTION
The production of a new analogy involves the construction of a new, often complex, relationship between two objects that was not previously part of the system's knowledge (Gick & Holyoak, Reference Gick and Holyoak1980; Gentner, Reference Gentner1983; Hofstadter, Reference Hofstadter2008). This new relationship, which we call an “association,” is then used to transfer knowledge from the source domain to that of the target (Gick & Holyoak, Reference Gick and Holyoak1983; Detterman & Sternberg, Reference Detterman and Sternberg1993; Barnden & Holyoak, Reference Barnden and Holyoak1994; Holyoak et al., Reference Holyoak, Novick and Melz1994; Gentner & Holyoak, Reference Gentner and Holyoak1997; Robertson, Reference Robertson2000). We concern ourselves with the former process, mapping, that is integral to both analogical reasoning and related processes such as conceptual blending (Fauconnier & Turner, Reference Fauconnier and Turner2003) and metaphor (Lakoff & Johnson, Reference Lakoff and Johnson2003).
Terminological precision in this field often suffers from the ability of many relevant words to refer to both processes and their results, so we adopt the following definitions throughout this paper. Mapping is the process of searching a source and a target object for a set of shared relationships on which to base a new analogy, while an association is the resulting set of relationships. Interpreting is the process of changing the representation of source and target objects to enable mapping, while a transformation is the resulting function that produces the new object representations from the prior ones. The processes of mapping and interpreting operate in parallel, iteratively generating candidate associations and transformations until a satisfactory pair of candidates, an association and the transformation that enables it, is found. Following structure mapping (Gentner, Reference Gentner1983), we adopt the notion that an association connects a set of relationships between features of one object's representation to an analogous set of relationships between another object's representation. Analogy depends on compatible relational representations (Doumas et al., Reference Doumas, Hummel and Sandhofer2008; Penn et al., Reference Penn, Holyoak and Povinelli2008; Holyoak, Reference Holyoak2012). Our focus is on how that representation can be reinterpreted during and because of the process of analogical mapping, and the effect of this reinterpreting on the resultant association.
Viewed with this focus, the process of mapping can be considered a search of the space of possible new relationships between two objects, where a new relationship is defined as a previously undiscovered commonality. This commonality can be a shared feature, a shared relationship between components, or any other matching representational structure. Due to the combinatorial nature of a space of possible mappings, the search can be computationally expensive, but conceptually the problem is simple: find matching pairs of representational elements. Many models of analogy that perform this kind of search exist, with ANALOGY (Evans, Reference Evans1964) being a very early example and the structure mapping engine (SME; Falkenhainer et al., Reference Falkenhainer, Forbus and Gentner1989) being the best known. These models and their computational implementations demonstrate that locating the shared properties between objects is a solved problem when compatible representations already exist.
The representation of an object that is conducive to a particular analogical mapping may not be familiar or typical. Contextually salient features may not be regarded as central to an object's description, nor might be the representational structure in which those features are placed. Analogical mapping does not necessarily concern the design attributes that are considered most salient to an object considered in isolation. The focus of this research is on how two objects with initially disparate representations can be reinterpreted in such a way as to be compatible for mapping. This parallels the distinction between constructing analogies and solving them identified in Harpaz-Itay et al. (Reference Harpaz-Itay, Kaniel and Ben-Amram2006). Previous approaches to this problem have focused on building representations from subsymbolic elements (Hofstadter & Mitchell, Reference Hofstadter, Mitchell, Holyoak and Barnden1992), iteratively constructing relational representations (Doumas et al., Reference Doumas, Hummel and Sandhofer2008), reconstructing representations from memory into the current context (Kokinov & Petrov, Reference Kokinov and Petrov2001), or by activating progressively more abstract concepts (Petkov et al., Reference Petkov, Vankov and Kokinov2011). Models of analogy making in design have focused on projecting representations into an ontology that characterizes their function(s), and then mapping based on patterns (Griffith et al., Reference Griffith, Nersessian and Goel1996; Qian & Gero, Reference Qian and Gero1996; Visser, Reference Visser1996; Bhatta & Goel, Reference Bhatta and Goel1997; Griffith et al., Reference Griffith, Nersessian and Goel2000), a special case of mapping by abstraction. This work seeks to define a general framework for how representations change during mapping that can both inform future computational models and serve to compare these existing approaches to representational transformation.
We apply this framework in the construction of a computational model of analogical mapping, Idiom, and describe several examples generated by an implementation of that model that could not have been constructed without interpretation-driven mapping (IDM). The implementation of Idiom is intended to serve as a proof of concept of both the framework and the model, demonstrating their computational feasibility and potential as an approach to analogy making. We then compare the way representations change in Idiom and three other analogy-making systems, demonstrating the differences of our approach.
2. THE IDM FRAMEWORK
Constructing representations suitable for analogy in a complex creative domain (such as design) is a complex task. At the heart of this complexity is a chicken-and-egg question: an association cannot be found without a representation that supports it, and yet whether a representation contains mappable elements cannot be determined without searching. Without representations that compatibly express the relations encapsulated within both objects, neither representation nor mapping can completely precede the other. We develop a framework for conceptualizing analogy in which these two processes take place interactively and in parallel. The central premise of our framework is that mapping should inform future representations, and representations should inform future mapping. Given symbolic representations of a source and target object, we model how those representations change over the course of the mapping process. We focus on this component of the analogy-making process to investigate how the context of searching for a mapping between source and target affects their representations. To facilitate this focus, we consider representation change during mapping separately from representation construction prior to mapping. The latter has long been considered a central component of analogy making and incorporated into models (Koestler, Reference Koestler1967; Wolstencroft, Reference Wolstencroft1989; Chalmers et al., Reference Chalmers, French and Hofstadter1992; French, Reference French2002); we focus on the former as a distinct but equally important consideration. This focus on representational change during mapping echoes the discussion on ontological mismatches in (Davies et al., Reference Davies, Goel and Nersessian2003).
We propose a framework, IDM, for this notion of representational transformation. In this formulation interpreting is dependent on candidate associations (i.e., how representations change depends on the progress of the search for mappings), and mapping is dependent on candidate transformations (i.e., how mappings are searched for is dependent on the progress of the search for representational changes). Our framework's name derives from this interaction: mapping is interpretation driven and vice versa. As a consequence of our desire for generality in this framework, we do not place any constraints on how the objects are represented, how their symbolic representations are initially constructed, what constitutes a satisfactory pair of candidates, or what happens to the association/transformation pair once it has been found.
Prior approaches to rerepresentation during analogical mapping include the minimal ascension method for extending structure mapping in Falkenhainer (Reference Falkenhainer1990), in which nonidentical relationships can be considered contextually mappable only when they have a common superordinate in the hierarchy of relationships. This is similar to the ontological approach used in Qian and Gero (Reference Qian and Gero1996), where relationships are mappable if they perform the same function, and to that used in McDermott (Reference McDermott1979) and Davies et al. (Reference Davies, Goel and Nersessian2003), where attributes are abstracted iteratively until they are mappable. Such approaches can be extended to leverage relational hierarchies from external sources like WordNet, as in Holyoak and Thagard (Reference Holyoak and Thagard1989), or from hierarchies inferred from large corpora, as in Turney (Reference Turney2008). Our iterative strategy for rerepresenting and mapping is similar to that adopted in Yan et al. (Reference Yan, Forbus and Gentner2003), although IDM's interpreting process is more general.
The IDM framework (Fig. 1) highlights the interactions between the rerepresentation of objects and the search for mappings. The perception process(es) that are precursors to the interpreting/mapping cycle, as well as the transfer process(es) that follow it, are included for illustrative purposes only: IDM makes no assumptions about those processes. We acknowledge transfer's crucial role in analogy making (Gick & Holyoak, Reference Gick and Holyoak1983; Hall, Reference Hall1989; French, Reference French2002), and its absence is a matter of scope, not of exclusion. IDM focuses on the cyclical relationship between mapping and interpreting.
The interpretation-driven mapping framework (center) embedded in a model of computational analogy. Interpretation-driven mapping addresses representation construction and parallelized search for mappings and object reinterpreting.
It has been argued (Chalmers et al., Reference Chalmers, French and Hofstadter1992; Kokinov & Petrov, Reference Kokinov and Petrov2001; French, Reference French2002) that the perception of analogs to be is not a precursor to the search for mappings but a concurrent process. The IDM framework supports and extends this notion, with representational change during mapping being its principal driver. While the perception process in Figure 1 operates prior to the processes of the framework, this sequential and discretized depiction allows us to focus on how representations change during the mapping process. We separate the processes acting to produce the representation of an object that exists when mapping begins (perception) from the processes that act to change the representation of that object during mapping (interpreting) based on their roles only, and make no assumptions about the similarity or difference in their mechanisms. We also refrain from stating that the symbolic representation that is present at the start of mapping is in any way “untransformed” or “canonical,” based on literature in embodied, constructivist, and situated cognition (Clancey, Reference Clancey1997; Schacter et al., Reference Schacter, Norman, Koutstaal and Bjorklund2000; Mahon & Caramazza, Reference Mahon and Caramazza2008) that disputes the existence of such Platonic conceptualizations. The initial representation is merely the representation existing at the time mapping begins, and the basis for representational transformations applied during that process.
The IDM framework does not prescribe analogy making completely; it is a framework for the interactions between representation and mapping, not a framework for the whole of computational analogy. The framework also does not make any assumptions about the nature of the mapping and interpreting processes or the representation of objects, transformations, and associations. This distinguishes it from previous approaches, such as the VAE model (Sowa & Majumdar, Reference Sowa and Majumdar2003), which utilizes a similar structure but limits possible transformations. The framework addresses the “which came first?” question in representation in analogy making by positing that the new relationship between objects and the representations that enable it arise out of a simultaneous, interactive process. Using the principles of IDM, it becomes possible to describe how any computational analogy-making model incorporates representational change into mapping. While other models were not derived from our framework, framing analogical representation in this way can enable scrutinization and comparison of otherwise disparate systems. The three core principles of the framework described, and the questions that arise from them when inspecting an analogy-making model through the lens of IDM, are as follows:
1. Mapping can be conceptualized as a search for an association between source and target. Constructing a new association is a search through a space of possible associations. Mapping is an iterative process that produces and evaluates candidate associations. From this two questions arise about a model:
• What bounds the space of valid associations constructible by mapping?
• What causes an association to be selected by mapping?
2. Interpreting can be conceptualized as a search for a transformation of source and target. Constructing representations that will be used in the association produced by mapping is a search through a space of possible transformations (to be applied to the initial object representations). Interpreting is an iterative process that produces and evaluates transformations. From this two questions arise about a model:
• What bounds the space of valid transformations constructible by interpreting?
• What causes a transformation to be selected by interpreting?
3. Interpreting occurs in parallel with mapping. The two processes operate simultaneously, with candidate transformations affecting mapping and candidate associations affecting interpreting. From this two questions arise about a model:
• How do candidate transformations selected by interpreting affect mapping?
• How do candidate associations selected by mapping affect interpreting?
3. IDIOM: A COMPUTATIONAL MODEL DERIVED FROM IDM
Idiom is a computational model of the association-construction component of analogy making based on the IDM framework. Idiom demonstrates the feasibility of the principles of parallel search and rerepresentation, and is so named for its ability to construct and operate on nonliteral representations of objects. The model describes one way representations can change during the construction of an analogy, and how those changes both affect and are affected by the search for mappings. Idiom provides one possible set of answers for the six questions raised at the end of Section 2.
IDM involves two bidirectionally interactive processes: mapping (or more generally the search for solutions) and interpreting (or more generally the reframing of the problem). The Idiom model adds a third process: perception, by which the representations comprising object features and the relationships between them are constructed from the sensory representations initially observed by the system. As an instantiation of the IDM framework, Idiom outputs a new association (a set of shared relationships between the source and target objects) and the representational transformation under which that association exists.
3.1. Model overview
We develop a symbolic representation of the three processes that comprise Idiom: perception, interpreting, and mapping (Fig. 2). Our model of representational change in analogy making begins with the model observing sensory representations of two objects. We do not treat the “source” and “target” objects differently because without transfer the two are interchangeable. We refer to these objects as “sensory representations” to emphasize that there is no veridical representation of any analog, only observations made through the system's perceptual faculties. The symbols used to represent the Idiom model are as follows:
a is a candidate association. An association a ∈ A is an ordered nonempty list of pairs of features, with the first of each pair being a node in τ(r(o 1)) and the second being a node in τ(r(o 2)).
a win is a candidate association that has been evaluated as satisfactory under the evaluation function
$\mathop {\longrightarrow}\limits^{q}$, leading to the Idiom model ceasing its search.A is the set of candidate associations being considered by the mapping process.
r(o 1), r(o 2) are the relational representations of the two objects that result from perception.
o 1, o 2 are the sensory representations of the two objects.
t is a candidate transformation.
t win is the transformation that was applied when a satisfactory association was discovered.
T is the set of candidate transformations being considered by the interpreting process.
τ is a candidate transformation that has been applied to the relational representations.
τ(r(o 1)), τ(r(o 2)) are the relational representations after the active transformation has been applied.
- $\mathop {\longrightarrow}\limits^i$ is the process of interpreting that uses the current association candidates to generate new ways to transform the object representations, and then selects one transformation to apply to them. See Section 3.4 for details.
- $\mathop {\longrightarrow}\limits^m$ is the process of mapping that searches the interpreted relational representations for associations based on structure mapping theory. See Section 3.3 for details.
- $\mathop {\longrightarrow}\limits^{p}$ is the process of perception that constructs relational representations from the sensory representations. See Section 3.2 for details.
- $\mathop {\longrightarrow}\limits^{q}$ is an evaluation function that can be applied to associations.
The model's perception process, 
$\mathop {\longrightarrow}\limits^{p}$, constructs the relational representations used in search, and can be represented as Equation (1). Relational representations consist of a set of features and a set of relationships between those features, and are a commonly adopted representation in analogy making (Gentner, Reference Gentner1983; Holyoak & Thagard, Reference Holyoak and Thagard1989; Hofstadter & Mitchell, Reference Hofstadter, Mitchell, Holyoak and Barnden1992; Kokinov & Petrov, Reference Kokinov and Petrov2001).
$$\mathop {\longrightarrow}\limits^{p} \; := {o_1} \longrightarrow r\lpar {{o_1}} \rpar \comma \; {o_2} \longrightarrow r\lpar {{o_2}} \rpar.$$The three processes of Idiom, showing the symbolic representations of their inputs and outputs. Idiom is a computational model that instantiates the interpretation-driven mapping framework.
The model's mapping process, 
$\mathop {\longrightarrow}\limits^{m}$, searches for associations between relational representations, and can be represented as Equation (2). As input it takes the transformed objects for searching and the set of candidate associations for updating. In Idiom the mapping process is dependent only on the current active transformation and its affects on the relational representations. This is a simplification of the IDM framework, in which mapping is reliant on the set of candidate transformations, allowing multiple transformations to influences mapping simultaneously, rather than a winner takes all approach used here. We denote the set of candidate associations after an iteration of the mapping process as A′ to indicate it has been updated. Idiom follows the structure mapping theory (Gentner, Reference Gentner1983) definition of analogical mapping, in which a mapping exists between a common subset of relationships between the features of the objects, although we reserve the word “mapping” for the process. As in Section 2, we refer to the shared subset of relationships between features that results from mapping as an “association.” We embed Gentner's theory in a system where the relational representations are iteratively transformed by an interpreting process in parallel with the search for mappings.
$${\mathop {\longrightarrow}\limits^{m}}\; := {\rm \tau} \lpar {r\lpar {{o_1}} \rpar } \rpar \comma \; {\rm \tau} \lpar {r\lpar {{o_2}} \rpar } \rpar \comma \; A \to a_{{\rm win}} {\vee} A^{\prime}.$$ The model's interpreting process, 
$\mathop {\longrightarrow}\limits^{i}$ , uses the set of candidate associations to construct new ways of transforming the object representations, and can be represented as Equation (3). As input it takes the object representations and the set of candidate associations for use in generating new transformations, and both the current active transformation and the set of candidate transformations for searching and evaluation. We denote the active transformation and set of transformations after an iteration of the interpreting process as τ′ and T′, respectively, to indicate they have been updated. Transformations act on the relational representations, and may include the addition or removal of features, or the addition, removal, or relabeling of relations. New transformations are generated based on what changes could improve the performance of associations in A according to the same success measure used to select candidate mappings 
$\mathop {\longrightarrow}\limits^{q}$. Transformations are generated, evaluated, and stored, with the most promising transformation at each point in time becoming the active transformation τ that is applied to the objects. This characterization of analogical mapping as being transformation dependent echoes the definition of similarity as “representational distortion” (Hahn et al., Reference Hahn, Chater and Richardson2003; Hodgetts et al., Reference Hodgetts, Hahn and Chater2009).
$${\mathop {\longrightarrow}\limits^{i}} := r\lpar {{o_1}} \rpar \comma \; r\lpar {{o_2}} \rpar \comma \; A\comma \; {\rm \tau} \comma \; T \longrightarrow T^{\prime}\comma \; {\rm \tau} ^{\prime}\comma \; {\rm \tau} ^{\prime}\lpar {r\lpar {{o_1}} \rpar } \rpar \comma \; {\rm \tau} ^{\prime}\lpar {r\lpar {{o_2}} \rpar } \rpar.$$ Associations can be evaluated by 
$\mathop {\longrightarrow}\limits^{q}$ under any transformation, allowing the system to iteratively improve its candidate associations based on the available transformations and its candidate transformations based on the available associations. A candidate association may specify unmappable feature pairs (those not connected within their respective objects under the current transformation). These invalid shared associations are ignored by 
$\mathop {\longrightarrow}\limits^{q}$.
Mapping and interpreting both operate incrementally, with each updated set of candidate associations A being the basis for an updated τ, which is then in turn the basis for an updated A, and so on. This iterative interaction implements the fundamental principles of the IDM framework. Below we describe the three processes in more detail.
3.2. Perception
The perception process in the Idiom model, 
$\mathop {\longrightarrow}\limits^{p}$, takes low-level sensory representations of a form appropriate for the domain in which it is implemented and constructs from them the features and relationships used in mapping. To construct relational representations, 
$\mathop {\longrightarrow}\limits^{p}$ extracts features from the sensory representations, clusters those features into concepts, and then constructs relationships between features based on their sensory and conceptual contexts. Details of the perception processes are particularly affected by the choice of an implementation domain, and in this model specification, we adopt an abstract framing to maintain broad applicability. The five processes that make up perception (Fig. 3) are the following:
• Feature detection identifies and describes elements of the sensory representation, which become the features of the object on which analogies will be based. Feature definitions are domain specific.
• Concept recognition classifies features into known conceptual categories based on a domain-specific similarity function.
• Concept generation constructs new conceptual categories from the features that were not able to be classified, which are then added to the set of known concepts.
• Featural relationship construction describes relationships between features based on the properties of those features.
• Conceptual relationship construction describes relations between features based on the concepts that they instantiate, for example, between two features that are instances of similar concepts.
The structure of the perception process of our model. The sensory representations are converted into featural, conceptual, and relational representations.
Feature detection acts on the sensory representations o 1 and o 2 to extract a set of features. A “feature” is a notion specific to the domain in which Idiom is implemented, and it may be any element of the object, either a discrete component or a description of some aspect of the whole. For example, in a musical domain, a feature might be a phrase of music, while in a visual domain, a feature might be a shape.
Each feature has a description embodying its properties, which is represented as a list of attribute/value pairs. The model requires that it be possible to calculate the similarity between two features for the purposes of categorization, but otherwise it makes no assumptions about their nature. The remainder of the perception process concerns the construction of the relationships between these features.
Concept recognition is one of the two processes involved in clustering the feature representations into conceptual groups for the purpose of inferring relationships between them. Concept recognition attempts to place the features extracted by feature detection into known categories, each a region in the space of possible features that we refer to as a concept. The concept recognition process determines which of the features observed in the objects belong to existing concepts. The set of known concepts persists over multiple iterations of the Idiom association-construction process. Features not matching an existing concept are passed to the concept generation process.
Concept generation is the second of the processes involved in classifying features by their conceptual categories. Those features that were not classifiable within the current conceptual hierarchy are placed into new concepts as needed. After both concept recognition and concept generation have operated on the feature representations, all features will have been tagged as members of at least one conceptual category. The conceptual and featural representations are then used in the construction of relational representations.
Featural relationship construction produces relationships between features based on their descriptions. A relationship consists of an ordered pair of features and a relation that connects them. This relation expresses some property of the second feature in the pair, the “destination feature,” in relation to the first feature, the “originating feature.” Relations are expressed relative to the originating feature to increase their generality for mapping. For example, “the destination feature is above the originating feature,” “the destination feature is twice the size of the originating feature,” or “the destination feature is a component of the originating feature.”
Conceptual relationship construction produces relationships between features based on the conceptual categories into which they have been placed. These express relations between the concepts two features instantiate, for example, “the originating feature instantiates the same concept as the destination feature,” “the originating feature instantiates a similar concept as the destination feature,” or “a concept instantiated by the originating feature is a parent concept of one instantiated by the destination feature.”
The resulting relational representations, consisting of a set of features and a set of relationships between then, are then searched based on structure mapping theory (Gentner, Reference Gentner1983): features in the source and target that share a pattern of relationships can become part of an association. Idiom performs this search while iteratively reinterpreting the object representations to enable otherwise-impossible associations.
3.3. Mapping
The mapping process in the Idiom model, 
$\mathop {\longrightarrow}\limits^{m}$, searches the representations provided by 
$\mathop {\longrightarrow}\limits^{p}$ in the context of the transformations applied by 
$\mathop {\longrightarrow}\limits^{i}$. Mapping and interpreting iteratively search and transform the relational representations until a suitable association is constructed. An individual iteration of the mapping process produces an updated set of candidate associations on which the next iteration of the interpreting process can operate. Mapping operates on the principles of structure mapping (Gentner, Reference Gentner1983), searching for sets of relationships between features within the individual objects that form shared patterns between the objects. For example, two objects could be associated by a pattern of features connected by a series of “immediately above” relationships. The two processes that make up 
$\mathop {\longrightarrow}\limits^{m}$ (Fig. 4) are the following:
• Association search generates new candidate associations based on shared relationships between the transformed source and target object representations. While there may not be a sufficient pattern of such relationships in the untransformed representations, the transformation acts to facilitate better associations.
• Association evaluation evaluates the patterns of relationships found by association search using the quality measure
$\mathop {\longrightarrow}\limits^{q}$. Each candidate receives an evaluation via that measure, with evaluations over a threshold resulting in an association being deemed suitable. If a suitable association is found, the process returns a solution, completing this run of the Idiom model. If no suitable mapping is found, the set of current candidates is returned for use by the interpreting process.• Association search discovers associations, sets of pairs of features that can be considered analogous, by searching for shared relationships in the transformed representations. This search applies not to the representations produced by perception, r(o 1) and r(o 2), but to their reinterpretations under τ. In the very first iteration of the mapping process, τ defaults to a null transformation in which the objects are unchanged, meaning that initially r(o 1) ⇔ τ(r(o 1)) and r(o 2) ⇔ τ(r(o 2)).
• Association evaluation determines the value of each candidate association as measured by
$\mathop {\longrightarrow}\limits^{q}$. While the specific measure is domain specific, structure mapping requires that relations must match exactly for them to be considered a valid component of the mapping. Here, $\mathop {\longrightarrow}\limits^{q}$ can evaluate mappings in a variety of ways, including measures based on the size (in features) of the mapped subset or measures based on the domain-dependent meaning of the features or relationships within that set.
Associations that map a pair of features within the source to a pair of features within the target must map at least one relation between them. For example, assume a group of features within one object are all the same size and all instantiate the same concept. Within the second object is a group of features that all share the same size, but instantiate different concepts. The association evaluation process would find an association between these two groups to be valid based on the shared size relationships alone. This is the advantage of reducing complex relations to a set of simple relations.
The structure of the mapping process of our model. The two transformed object representations are searched for associations that fit the evaluation measure.
The mapping process updates the set of candidates. If a candidate association a scores sufficiently highly at 
$\mathop {\longrightarrow}\limits^{q}$ then the Idiom model terminates, returning a and τ. Otherwise 
$\mathop {\longrightarrow}\limits^{i}$ is executed and the cycle of searching and transforming continues.
3.4. Interpreting
The interpreting process in the Idiom model, 
$\mathop{\longrightarrow}\limits^{i}$, uses the current set of association candidates produced by the last iteration of 
$\mathop {\longrightarrow}\limits^{m}$ to determine a new active transformation from among the candidates. To do this 
$\mathop{\longrightarrow}\limits^{i}$ constructs, evaluates, and applies transformations.
The three processes that make up 
$\mathop {\longrightarrow}\limits^{i}$ (Fig. 5) are the following:
• Transformation search constructs new transformations by searching the current set of candidate associations for what could make them successful.
• Transformation evaluation considers all transformations in the current set of candidate transformations against the candidate associations to determine which should be the active transformation τ.
• Transformation application applies the newly updated active transformation τ′ to the relational representations of the source and target objects, reinterpreting them and affecting the next iteration of the mapping process.
The structure of the interpreting process of the Idiom model. Transformations are generated from candidate associations. One is selected as the active transformation τ and applied to the object representations.
Transformation search is the process by which new transformations used in interpreting are constructed. A generate and test approach is used to create transformations that would make associations in the current candidate set more successful at mapping between the two objects. Transformations are applied to the representations being searched by 
$\mathop {\longrightarrow}\limits^{m}$. Each transformation must be general enough to be applied to any possible relational representation. Possibilities include the deletion, addition, editing, consolidation, or splitting of both features and relations. A transformation may affect the relational representations by editing and rerunning the perception process to produce altered representations.
Because the purpose of interpreting is to render nonmatching relationships matching, the simplest transformation(s) involve the editing of relations. An example of such relabeling would be a “spatial reversal” transformation, under which all relationships involving “in front of” were treated as equivalent to “behind,” and so on. This transformation would parallel the “opposites” slippage in Copycat (Hofstadter & Mitchell, Reference Hofstadter, Mitchell, Holyoak and Barnden1992) that enabled analogies such as “ABC : ABD is like XYZ : WYZ.” An alternative approach to defining transformations is the adjacent node combination of the VAE model of analogical reasoning (Sowa & Majumdar, Reference Sowa and Majumdar2003). Processes by which transformations can be constructed include attempts to improve current highly performing associations according to 
$\mathop {\longrightarrow}\limits^{\rm q}$, or recalling transformations that have been successful on previous analogy problems.
Transformation evaluation produces an evaluation of all the candidate transformations in T by applying them to the candidate associations in A and measuring their quality with 
$\mathop {\longrightarrow}\limits^{q}$. Given the computational complexity of this, a sampling approach is likely warranted, involving selecting a subset of “test” candidate associations from A for each candidate transformation in T. This selection could be weighted by 
$\mathop {\longrightarrow}\limits^{q}$(a). Once an evaluation of the quality of each transformation in T is known, one candidate becomes τ and influences mapping.
Transformation application applies the current τ (either new or retained from the previous increment) to the representations r(o 1) and r(o 2). After the new transformation is applied, this increment of the interpreting process completes and mapping continues.
4. CONSTRUCTING ASSOCIATIONS WITH IDIOM
We have developed an implementation of the Idiom model that demonstrates its feasibility and serves as a proof of concept of the IDM framework. We present selected results from the implementation showing its capacity for interpreting and mapping.
4.1. Implementing Idiom
The implementation of Idiom described in this paper is in the domain of visual analogy, constructing relationships between images and drawings based on their spatial properties. The Idiom model is domain agnostic, but this implementation serves as a proof of concept of both Idiom and the IDM framework from which it is derived. Associations constructed by this implementation will be based on perceptual relationships, such as geometrical or spatial relationships between shapes. This system can perform mapping on shapes similar to those of CogSketch (Lovett, Gentner, et al., Reference Lovett, Gentner, Forbus and Sagi2009; Lovett, Tomai, et al., Reference Lovett, Tomai, Forbus and Usher2009; Forbus et al., Reference Forbus, Usher, Lovett, Lockwood and Wetzel2011), but our focus is on representational transformation and the integration of representation and mapping, rather than on relational abstraction. Visual associations are an important component of many design processes in many design domains.
The implementation takes visual representations of objects in the form of two-dimensional vector drawings, extracts features from them, constructs relationships between those features, and then makes analogies using those relationships. The definition of a “feature” in this implementation is a minimal closed shape, and features are described using the outline of these shapes, not their weight, colors, or other styling. These shape features are categorized into groups based on the similarity of their outlines using a centroid-radii method for shape description proposed by Tan et al. (Reference Tan, Ooi and Thiang2003). The shape-based representations are analyzed for a variety of featural relationships, including relative scale, orientation, position, and the sharing of edges and vertices. Conceptual relationships include conceptual similarity and conceptual identity. Featural relationships are abstracted into discrete categories like “slightly smaller” or “10° difference in orientation.”
Mapping is computationally modeled as an edge-labeled subgraph isomorphism search, with each feature being a node and each relation an edge. An exact maximum subgraph isomorphism search is computationally intractable (Garey & Johnson, Reference Garey and Johnson1979; Kann, Reference Kann1992), but here approximate solutions are acceptable. We adopt a genetic algorithm-based subgraph isomorphism approximation based on Wang and Zhou (Reference Wang and Zhou1997) and Cross et al. (Reference Cross, Wilson and Hancock1996), in which the genotype is a set of node–node connections between object graphs, and the fitness function is based on the number of connections that are valid (our quality measure). Transformations applied to the object graphs change the fitness landscape by affecting the validity of these mappings. A population of candidate associations between the source and target are created randomly and then evolved, with the quality measure 
$\mathop {\longrightarrow}\limits^{q}$ acting as the fitness and the current τ applying to all evaluations. The quality measure is based on the size of the largest contiguous subgraph validly mapped (i.e., the number of features connected by a pattern of shared relationships in the analogy). Mapping size is only an approximation of analogical quality, but its simplicity permits us to focus on interpretation.
Transformations in this implementation select one or more pairs of relations and treat them as interchangeable for the purpose of mapping. This enables interpreting of the kind “treat X in the source object as if it were the same as Y in the target object.” This gives more interpretive freedom than the approach used in Galatea (Davies & Goel, Reference Davies and Goel2001; Davies et al., Reference Davies, Goel and Nersessian2003; Davies & Goel, Reference Davies and Goel2003), which can only draw from a set of prespecified transformations. Selection of transformations is based on the question “what transformation would improve the mappability of these two objects the most?” This relies on underlying syntactic commonality rather than any semantic content. When two patterns of relationship share the same structure, but not the same relationships, a transformation can be constructed to treat them as mappable. For example, an object containing a line of shapes of increasing sizes could be associated with another object containing a line of shapes of increasing orientations. The approach is related to the corpus-based label substitution in EMMA (Ramscar & Yarlett, Reference Ramscar and Yarlett2003), where similarity along the dimensions that best describe a large data set of examples is used to substitute contextually similar word. Idiom, by contrast, seeks out transformations that can most improve the mappability of the source and target object.
The Idiom approach expands the space of possible transformations compared to previous systems, allowing a greater diversity of mappings. It does so at the cost of explicability: transformations are justified based on their effect (enabling a more complete mapping) rather than their meaning (the relationship between the concepts so transformed). Section 5 compares Idiom's capabilities with other analogy systems.
4.2. Example analogies in visual art and ornamental design
A variety of vector representations of visual art objects, ornamental design motifs, and architectural objects were drawn from images, taking care to select objects that could be meaningfully represented by lines and shapes. Several of the resulting associations are presented here, along with descriptions of how they were produced. We demonstrate that without IDM, a structure-mapping based system could only have constructed these analogies with task-specific representations.
4.2.1. Example 1
Figure 6 shows the first example of the output of the computational implementation of Idiom. The figure depicts the vector representations of Object 1 (o 1), a Hittite sun symbol, and Object 2 (o 2), a French empire motif, both from Humbert (Reference Humbert1970). The thick, gray and black solid lines indicate the lines that make up the object representation itself, with the black lines indicating object features that are part of the association. The thin solid lines connecting features in Object 1 with features in Object 2 indicate the pairs of features that have been mapped by the successful association a win. The thick black dashed lines joining features within each object indicate the relationships between those features that form the basis for the mapped relationship. Each of these connections is labeled with the relationship it represents. The gray box at the bottom of the figure contains the current value for the active transformation τ.
An association produced by Idiom that maps adjacent points of the seven-pointed star in Object 1 to adjacent petals of the top floret in Object 2, based on interpreting the “50° orientation difference” relationships between the star points to be like the “20° orientation difference” relationships between the petals.
In Figure 6, the seven points of the star in o 1 are mapped to the seven petals of one floret in o 1, with adjacent points being mapped to adjacent petals. Each of the adjacent pairs of points in Object 1 are connected by an “approximately 50° difference in orientation” relationship (abbreviated using △rot in the diagrams), while each of the adjacent pairs of petals in Object 2 are connected by an “approximately 20° difference in orientation” relationship. The association was made in the presence of a transformation, τ, equating these two relationships. Idiom is making the analogy between the star and the floret in the context of treating those two relationships as alike. Idiom possesses no knowledge about the relative similarity or difference of the two orientation relationships, and they could just as easily have been one orientation relationship and one of scale, or any other relational type possessed by Idiom. The transformation that enables this mapping was selected on the syntactic similarity of the representations of the two objects, which each possessed a structurally similar pattern of homogeneously labeled edges. Naive structure mapping would have failed to construct this association without additional knowledge.
Idiom's process to construct the association seen in Figure 6 began with the system attempting to map between the untransformed representations. The process of mapping, 
$\mathop {\longrightarrow}\limits^{m}$, creates (initially by chance) a candidate association, a 1, that maps at least one pair of adjacent star-points to a pair of adjacent petals. Association a 1 is not of any value by the quality measure 
$\mathop {\longrightarrow}\limits^{q}$ because there is no suitable transformation available to equate the two different orientation labels. The process of interpreting, 
$\mathop {\longrightarrow}\limits^{i}$, then works backward to construct a transformation that would increase the value of a 1, and one of the candidate transformations thus constructed is the one shown in Figure 6, which we will call t 1. Because features can be mapped using t 1 that cannot be mapped without it, it outperforms other available transformations, including the default null transformation in which the representations are unchanged, and becomes the active transformation τ. This transforms the representations to treat the 20° and 50° orientation relations as alike. The mapping process 
$\mathop {\longrightarrow}\limits^{m}$ then adopts a search trajectory to maximize the number of associated nodes given τ = t 1. This produces an association like the one in Figure 6.
Theoretically, if the objects were represented in precisely the right way, the association depicted in Figure 6 could be constructed without the use of representational transformations. A relationship like “rotationally adjacent” or “sharing a common axis of rotation” could produce mappable representations of Objects 1 and 2 for which no transformational process was needed. Alternatively, relationships between relationships, such as those used in some structure mapping theory based systems (Falkenhainer et al., Reference Falkenhainer, Forbus and Gentner1989; Forbus et al., Reference Forbus, Gentner and Law1995), could be used to describe the orientation differences in a more general way, and these second-order relationships would be mappable. However, these approaches require significant prior knowledge on behalf of the experimenters to construct representations of the two objects that feature compatible relationships. The Idiom method discovers appropriate transformations from regularity in the structure of the object representations, rather than from compatible relational labels.
Idiom can produce different results when run repeatedly with the same stimuli. Figure 7 shows the result of another run involving the same two objects. In this case, six of the seven points of the star in Object 1 are mapped to the two outermost petals of each floret in Object 2. The difference in orientation between every third star point (150°) is associated with the difference in orientation across each floret and between each floret, which are both 120°. The analogy-making system has interpreted these two different relationships as being alike, transforming different parts of the object representations to the first example (Fig. 6) and producing a different analogy.
A different association between the same two objects, in which every third point in the seven-pointed star in Object 1 is mapped to the set of outermost petals on the three florets in Object 2.
These examples demonstrate that the Idiom model can produce analogies between simple visual objects. The computational implementation of this model has produced multiple varied outputs depicting associations that are different in both appearance and substance. It does this by “forcing” mappings between semantically different relationships based on the underlying syntactic similarities of the objects being mapped, such as the patterns of 120° and 150° relationships transformed in Example 2 and the similar pattern transformed in Example 1.
4.2.2. Example 2
Among the vector representations of art and design objects provided to the implementation were a line-art version of M. C. Escher's 1938 painting “Sky and Water I” and a 19th-century wrought ironwork pattern intended for use in a gate (Cottingham, Reference Cottingham1824, via Cliff, Reference Cliff1998). An association between these two objects can be seen in Figure 8, where the four rows of fish that make up the bottom half of Object 3 have been mapped to the four rows of quadrilateral-like shapes in the middle of Object 4. The features of each row of fish belong to their own conceptual category, as the fish become progressively more abstract toward the center of the painting, resulting in a ”similar concept” relationship existing between adjacent rows. The curved bars that make up the ironwork pattern in Object 4 create a similar pattern, with the gaps between the bars becoming slightly smaller with each successive row. The Idiom implementation uses a transformation that equates these two relationships and maps between the fish and the gaps.
An association produced using Idiom that maps the triangular pattern of increasingly more abstract fish in Object 3 to the triangular pattern of increasingly smaller gaps in the gate in Object 4, based on interpreting the “similar concept” relationship between the fish to be like the “slightly smaller” relationship between the gaps.
Figure 8 shows that Idiom can construct relationships between structurally similar representations even when the relationships involved are disparate and the objects being represented are in different domains. Not only could naive structure mapping not have constructed this association without additional knowledge, but also it is difficult to conceptualize a representation in which the relationships of similarity and scale are expressed identically. The two relationships are objectively different, but in the context of this association, it is of potential benefit to explore what the world would look like if they were not. Using IDM, our system is capable of mapping a pattern of typological relationships between figures in a painting to a pattern of topological relationships between figures in an ironwork gate. This result demonstrates the possibility of interdomain association using Idiom given symbolic representations of both objects.
5. COMPARING REREPRESENTATION IN ANALOGY MODELS
In addition to serving as a basis for developing models of computational analogy making, the IDM framework can serve as a lens through which the interpreting capabilities of analogy-making systems can be described. The constituent processes of models of analogy can themselves be analogously mapped to IDM's perception, interpreting, and mapping. An analogy-making model can be described using IDM so long as it exhibits a mechanism by which previously unmappable object representations can be transformed so as to make them mappable. By reframing other systems through this lens, we can describe commonalities in the way they transform representations despite the diversity of methods by which they do so. The IDM framework asks six questions of an analogy-making model (see Section 2), and we follow this line of questioning for three extant models and the Idiom model described in this paper.
The three models used for comparison come from the three paradigms of computational analogy-making models in French (Reference French2002): the symbolic, the connectionist, and the hybrid. The canonical examples of each (according to French) are the SME (Falkenhainer et al., Reference Falkenhainer, Forbus and Gentner1986, Reference Falkenhainer, Forbus and Gentner1989), analogical mapping by constraint satisfaction (ACME; Holyoak & Thagard, Reference Holyoak and Thagard1989; Spellman & Holyoak, Reference Spellman and Holyoak1992), and Copycat (Hofstadter, Reference Hofstadter1984; Hofstadter & Mitchell, Reference Hofstadter, Mitchell, Holyoak and Barnden1992). However, the SME is a model of analogical mapping only and does not incorporate any representation construction or transformation. We compare an extension of it, incremental structure mapping (I-SME) (Forbus et al., Reference Forbus, Ferguson and Gentner1994), which incorporates a form of reinterpretation into the basic SME model. The incremental nature of I-SME addresses the problem of cognitive plausibility when problem solvers are exposed to knowledge about potential analogues sequentially, with new knowledge arriving after the mapping process has begun. I-SME adds new knowledge to object representations during mapping, and decides in each case whether to extend the existing mapping or construct a new and different one. We compare I-SME, ACME, Copycat, and our Idiom implementation through the lens of the IDM framework. I-SME is a general theory of analogical mapping, and thus can map between any symbolic representation, but the best-known implementation of it, the minimal analogical reasoning system, solves engineering problems given a worked example of an analogous problem. This implementation of I-SME combines basic knowledge about objects, incremental mapping, and an equation solver to solve problems. It incrementally extends its representations by generating candidate inferences, evaluating them, and incorporating the successful ones into its mapping.
ACME maps between symbolic representations of real-world concepts, and is able to construct analogies like “Socrates is a midwife of ideas.” ACME operates on graph representations of features and relationships, and constructs mappings by spreading activation through that graph, with simultaneously active concepts being analogous. This activation is guided by a set of constraints governing what kinds of properties should be mapped. ACME adds excitatory and inhibitory connections between features to constrain mappings between those features to be similar in structure, similar in meaning, and relevant to the current context.
Copycat finds analogical connections within the domain of letter strings, solving proportional analogies of the form abc → abd, ijk → ? Copycat has knowledge of concepts relevant to the domain, such as group of letters, identical letters, and successive letters, in the form of a connectionist semantic network. Copycat builds representations of letter strings out of these concepts using a parallel representation construction and mapping search process. Copycat's search is affected by both top-down pressures from the conceptual network and bottom-up pressures from the letter strings.
Our implementation of the Idiom model (see Section 4.1) finds analogous patterns of shapes within vector images drawn from ornamental design, architecture, and art. Images represented relationally are iteratively searched for mappings by a genetic algorithm and transformed by an interpreting process driven by reinforcement learning. We demonstrate that the IDM framework is sufficiently general as to compare the mechanisms of each system. The first question posed in the IDM framework is “What bounds the space of valid associations constructible by mapping?” (Table 1). The mapping search spaces used by I-SME, ACME, Copycat, and Idiom all share a commonality: they map only between features that are contextually identical. They differ in their representations, search strategies, and definitions of “context,” but each will map patterns of matching features. Copycat, I-SME, and Idiom follow structure mapping and map between relationships, not features. ACME maps features, but its process for doing so can be thought of as contextually equating the relationships between them as identicality, and thus can also be thought of as mapping structures of relationships.
Defining the space of valid associations
The second question posed in the IDM framework is “What causes an association to be selected by mapping?” (Table 2). The four models can be divided into two categories: those that implement transfer and those that do not. The models that include a transfer component, I-SME and Copycat, terminate when they can successfully transfer knowledge from the source to the target and solve the provided problem. ACME terminates based on local maxima, by reaching a state where no further mapping can occur. Idiom terminates based on a specified suitability criterion that is specified according to the problem domain. The Idiom implementation presented in this paper uses a measure of quality derived from the association structure (the mapping's size in nodes).
Selecting candidate associations
The third question posed in the IDM framework is “What bounds the space of valid transformations constructible by interpreting?” (Table 3). Three of the four systems, ACME, Copycat, and Idiom, implement transformation as contextual equivalency between relationships. They differ, however, in how and why these transformations are generated. In ACME and Copycat, relational equivalencies are enabled by prespecified connections between concepts, although in both this affects representations indirectly. In ACME this enabling connection is in the form of constraints favoring certain mappings over others, and in Copycat certain concepts are “slippage” capable in specified circumstances. The connectionist nature of ACME means that these equivalencies are specified in a distributed but deterministic fashion, while the hybrid connectionist/symbolic nature of Copycat means that equivalencies are symbolic but only affect representations through distributed processes. In Idiom the interpreting process is symbolic and stochastic as in Copycat, but all pairwise equivalencies are possible as in ACME.
Defining the space of valid transformations constructible by interpreting
I-SME differs from the other three systems in that transformation does not take the form of contextual equivalency between features of the object representations. I-SME can receive new knowledge between mapping steps either from an external source like the experimenter or through its own deduction or inference. These transformations do not contextually treat objectively different properties as contextually similar, but instead represent an evolving objective knowledge base. For example, I-SME could be implemented with a set of knowledge about cats (that they are kept as pets by humans, that they have fur, etc.) and use that to add knowledge to an object once it had established that the object belonged to the class “cat.” Transforming between levels of representational abstraction like this is not the same as contextual equivalency between objects: a cat is not a pet only for the purposes of this mapping; it is always both a pet and a cat.
The fourth question posed in the IDM framework is “What causes a transformation to be selected by the interpreting process?” (Table 4). Two of the four analogy-making models, I-SME and Idiom, select transformations based on how they improve the associations they act on, while the other two, ACME and Copycat, use contextual influence. I-SME's selected transformations (the inferred attributes that extend its representations) improve candidate associations based on progress toward the goal state, while Idiom's selected transformations improve candidate associations based on the measure of association quality. Both I-SME and Idiom use syntactic properties of the object representations to motivate transformations: the transformations that are generated make candidate associations stronger according to each system's goal. Because Idiom can construct any pairwise equivalency between relationships, this means it is driven by structural similarity within object relationships, and shared patterns of different relationships that can be equated by transformations. I-SME, by contrast, is limited in its transformations by what it can derive from existing knowledge using rules and inference, and cannot contextually change the meaning of its representations, only add to them.
Selecting candidate transformations by the interpreting process
ACME and Copycat select their transformations using the effect of the current task context. In ACME, every node exerts influence, however small, on every other node's activation and thus what it is analogous to. In Copycat, representational elements beget more like them, but there is also influence exerted by active concepts, and active concepts spread their activation to similar concepts in the same way as ACME. Copycat's transformations (referred to as “conceptual slippage”) occur only in the right combination of conceptual activations, in contrast to ACME's transformations, which occur as the sole, iterative mechanism by which analogies are constructed.
The fifth question posed in the IDM framework is “How do candidate transformations selected by interpreting affect mapping?” (Table 5). While the cause and scope of their transformations are very different, I-SME and Idiom both use transformations that directly change the representations being searched. This affects the search trajectory by redefining the problem space: what was once mappable may now not be, and vice versa.
Effects of candidate transformations on mapping
ACME and Copycat, while they both use spreading activation among conceptual networks to trigger their transformations, have very different processes for how those transformations affect mapping. In ACME, the network of nodes and relationships between them is the only representational structure present, and transforming the meanings of those nodes (expressed as the context of simultaneous activations) is both the mapping process and the interpreting one. This equivalence between transformation and mapping reflects the core of the connectionist approach: “meaning” exists only in context, and thus both transformations of meaning and solutions to problems are embedded in distributed structures. While this may seem to violate a comparison of this system with the others under the IDM framework, note that it is only the last two questions that are difficult to answer in ACME, and their answering still proves informative.
The sixth and final question posed in the IDM framework is “How do candidate associations selected by mapping affect interpreting?” (Table 6). Each of the four systems provides a very different answer to this question. In I-SME, the rules of symbolic logic apply: transformations can only occur when there is a clear logical precedent. In ACME, again, the question does not have a clear meaning, because transformations and association candidates are both expressed as conceptual activation patterns. In Copycat, there are indirect effects on transformation brought about by the current candidate associations: existing representational fragments affect conceptual activation, and certain patterns of conceptual activation trigger transformation. In Idiom, the relationship is more direct: candidate transformations are evaluated based on how they improve candidate associations, meaning that the evaluation of the former is contingent on the state of the latter.
Effects of candidate associations on interpreting
This comparison illustrates that the IDM framework serves as a useful lens through which to view analogy-making models and compare the ways by which their constituent processes interact. IDM is an effective comparative tool for models of analogy that have historically been categorized as divergent along the “symbolic–hybrid–connectionist” axis. The comparison also shows that Idiom occupies a niche distinct from previous models of analogy making: it combines the structure mapping approach of I-SME with the contextual rerepresentation of Copycat while retaining the flexibility of ACME.
6. CONCLUSION
The way representations change during mapping is a critical component of computational models of analogy. We have developed IDM, a general framework for describing representational transformation within analogy making. IDM focuses on the way iterative reinterpreting occurs in parallel with the mapping process, and how this process mutually affects and is affected by mapping. This general framework has the capacity to investigate how other models of analogy making have incorporated these interactions. By representing other analogy-making models in terms of these mechanisms, we are able to elucidate the similarities and differences in the way their object representations change during mapping. We argue that this framework is particularly apt for analogy in computational design, because the situated, constructive, and emergent nature of design representations (Gero, Reference Gero1998) requires an approach that supports representational fluidity.
IDM provides a unifying descriptive architecture for comparing the processes used by diverse computational analogy-making models. Reviews of computational analogy making spanning the last 25 years (Hall, Reference Hall1989; French, Reference French2002; Gentner & Forbus, Reference Gentner and Forbus2011) have identified a number of typicalities in extant models. These include typical processes in models of analogy (including retrieval, rerepresentation, mapping, transfer, and evaluation), typical components of the representations used in analogy (such as labeled relational structures), and typologies of model architectures (connectionist, symbolic, and hybrid). The IDM framework introduces an additional, general, way to describe similarities among analogy-making models: their interpretive capacity and its catalyst(s). The IDM framework then poses questions about the representational processes encapsulated in the model and their interactions with mapping. The answers to these questions provide new insight into how models previously thought of as theoretically incompatible, such as connectionist ACME and symbolic I-SME, are alike.
We have developed a computational model of the mapping component of analogy making, Idiom, that instantiates the IDM framework. Idiom is specifically designed around the iterative and parallel interactions between rerepresentation and the mapping process, and serves as a proof of concept of the IDM framework. Idiom, and the prototypical implementation of it in the domain of ornamental design, demonstrates that a focus on the changes in representation that occur during mapping can lead to an expanded capacity for mapping without requiring extensive or specially built representations. Idiom leverages structural similarities that underlie object representations to transform the relational structures and enable mapping. Two objects that may not share any relations may share a pattern of how those relations occur between their features, and it is from this syntactic level of commonality that Idiom derives its transformations.
We evaluate our model by its generative capability: it can construct associations that previous structure-mapping systems could not construct from the same representations. This capability stems from the parallel interactive interpreting and mapping of the IDM framework. The model is able to construct a variety of associations from the same pair of objects by using different transformations, and able to construct associations between relationships regardless of their similarity, permitting cross-domain analogy. We make no claims as to the quality of the associations produced (beyond the internal evaluation by Idiom of the number of features they can connect), nor of their utility for a particular task: Idiom is a model only of the mapping component of analogy making and produces associations in the absence of transfer. Idiom and IDM framework provide capabilities that structure-mapping systems did not previously possess. Their extension to a full model of analogy including transfer and evaluation is an area of active research.
To illustrate the capabilities of both the IDM framework and Idiom, we compare our implementation to three analogy-making models, I-SME, ACME and Copycat. Symbolic models like I-SME typically utilize logical inference as the mechanism for representational change when transformation during mapping is possible at all. Idiom's approach offers more flexibility than the I-SME approach in both the construction and the selection of transformations: new transformations can be constructed from syntactical regularities in object representations, and the iterative stochastic transformation evaluation process is less constrained than inference (at least in the absence of new external knowledge). The approaches to transformation used by ACME and Copycat are revealed by the IDM reframing as diametrically opposed: while both model transformations as contextual equivalency, Copycat has few potential transformations (“slippages”), while ACME allows any pair of features to transform. Conversely, Copycat's slippages occur only after specific representational and conceptual states have been reached, while ACME's transformations occur constantly as a result of aggregate pressure from the entire representational structure. Idiom, by comparison, permits the same space of possible transformations as ACME (as in theory any two relations can be equated) while offering guidance in selecting transformations as in Copycat (as transformations are only constructed in the context of specific, favorable representational states). This shows the strength of the iterative and parallel approach to interpreting and mapping: everything can in theory be mapped, but local conditions guide individual decisions about which transformations to apply.
The second strength of the IDM framework is in the centrality of the interactions between mapping and interpreting. Interpreting in Idiom affects mapping by transforming the object representations directly, changing the trajectory of the search as in I-SME and ACME. Mapping in Idiom affects interpreting by driving what transformations are constructed: the interpreting process explores how to improve extant candidate associations in the same way that in Copycat conceptual slippage occurs when context suggests it would be useful. Unlike Copycat, Idiom is domain general, and its transformations are syntactically derived, obviating the need to specify a domain- and context-specific set of potential slippages. The IDM approach allows mapping and interpreting to iteratively influence each other in a domain-general model of analogy.
This comparison demonstrates the descriptive power of the IDM framework and highlights that representational variation during mapping is central to analogy. Idiom exhibits capabilities previously only observed separately in models of analogy making, specifically the ability to contextually guide the selection of a representational change and the ability for a large space of possible representational changes to be applied. These characteristics, which are made possible by Idiom's ability to learn new transformations based on the structure rather than the content of the objects it is mapping, make Idiom particularly suited to the representationally dynamic domain of analogical reasoning in design.
Kazjon Grace is a Postdoctoral Researcher at the University of North Carolina at Charlotte. He completed his PhD at the University of Sydney in 2012. His research investigates computational creativity and its applications to supporting design and incorporates techniques from artificial intelligence, interaction design, machine learning, and design cognition. Dr. Grace is motivated by the development of intelligent creative systems that can be active partners alongside human innovators. His work includes systems that reinterpret objects in order to solve problems, systems that can recognize creative products, and systems that harness crowd creativity to improve themselves.
John Gero is a Research Professor in computer science and architecture at the University of North Carolina at Charlotte and in Computational Social Science at the Krasnow Institute for Advanced Study, George Mason University. He has been a Visiting Professor of architecture, civil engineering, cognitive psychology, computer science, design and computation, and mechanical engineering at MIT, University of California–Berkeley, University of California–Los Angeles, Carnegie–Mellon University, and Columbia University in the United States and in the United Kingdom, France, and Switzerland. His research spans design computing and design cognition that is and has been supported by the NSF, DARPA, and NASA. Dr. Gero is the author or editor of 52 books and over 650 published papers.
Rob Saunders is Head of the Design Lab at the Faculty of Architecture, Design and Planning, University of Sydney. Rob's research explores the role of curiosity in individual, social, and cultural creative systems. Using techniques from artificial intelligence, machine learning and robotics, he develops computational models of curiosity, artificial creative systems, and creative robotics. Dr. Saunders works with artists and designers across a range of disciplines to support and engage in the creative application of computing and has applied his research in the development of design customization systems, smart environments, interactive installations, and robotic artworks.
